Operator Help

im trying to show that if we know that an operator U acts on a vector a we get a resulting vector b

Ua=b

im trying to show that because we know this we know that the vector component of b, bi, is equal to the sum of the operator matrix, Oij, multiplied by the vector component of a, ai

bi = Sum(Oij aj)

so i have no idea how to post equations on here

so far im assuming that we need to utilize the kroneckers delta to show that the basis of these vectors are othanormal to each other, and therefor when multiplied either equal 0 or 1.

So some things im not sure of is if we can assume that the two vectors, a and b, occupy the same space and therefor have the same basis.

any help at all would be lovely, and if you need clarification id be more then happy to help, this is kinda vague

scratch that, its really easy i was just being stupid

if anyone got this question to start, then answer is that we can write the transformed vector a into
Uai=sum(Oijaj), basically its saying that when you are multipling a vector by an operator, you get out the sum of operation in matrix formed multiplied by a vector orthogonal to ai, which in this case was b!